Timeline for Euler characteristics with and without compact support of algebraic varieties
Current License: CC BY-SA 3.0
16 events
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| Jun 16, 2014 at 14:31 | comment | added | Reladenine Vakalwe | @semyonalesker: Thanks for pointing me to the Laumon reference! I should have looked at it to start with. I was missing one key argument in my proof for $[f_*] = [f_!]$. Turns out there is something in Laumon's paper that I can adapt that finishes the job. | |
| Jun 16, 2014 at 7:12 | vote | accept | asv | ||
| Jun 16, 2014 at 7:08 | comment | added | asv | @ReladenineVakalwe: Regarding your "Added even later" remark on maps $[f_*],[f_!]\colon K_0(X)\to K_0(Y)$. This was shown by Laumon (1981) for $l$-adic sheaves in the paper quoted by Kestutis Cesnavicius in his answer below, including the positive characteristic case. I do not know whether this formally implies the corresponding statement for sheaves with $\mathbb{C}$-coefficients on complex varieties. | |
| Jun 12, 2014 at 13:34 | history | edited | Reladenine Vakalwe | CC BY-SA 3.0 |
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| Jun 12, 2014 at 13:19 | history | edited | Reladenine Vakalwe | CC BY-SA 3.0 |
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| Jun 12, 2014 at 12:36 | history | edited | Reladenine Vakalwe | CC BY-SA 3.0 |
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| Jun 12, 2014 at 2:26 | comment | added | Reladenine Vakalwe | Perhaps, the relevant point here is that the Euler characteristic of the link of the closed stratum is $0$. In fact, I think a variant of this argument yields that the Euler characteristic and compactly supported Euler characteristic with coefficients in any algebraic constructible complex of sheaves on $X$ coincide. This latter statement might actually even be an easier statement to prove (I am thinking along the lines of the proof of Artin vanishing that uses induction on dimension, but it's too late at night for me to think through this properly right now). | |
| Jun 11, 2014 at 17:33 | comment | added | Reladenine Vakalwe | You are of course right (I was too hasty with my comment picturing only smooth points in my head, sorry). However, the Whitney conditions on the strata yield more control on the local picture at each point of the stratum. Instead of trying to explain myself, let me point to notes of MacPherson: faculty.tcu.edu/gfriedman/notes/ih.pdf, see Theorem 7.2 and 7.3 on page 138. I hope that clears things up? | |
| Jun 11, 2014 at 14:21 | comment | added | asv | @ReladenineVakalwe: Still I do not see how you get the sphere. Assume for simplicity that $Y$ is a point. Then the tubular neighborhood of $Y$ does not have to be a ball if $X$ is singular at the point $Y$, if I understand correctly. Then if you remove $Y$, you may not get something homotopically equivalent to the sphere. But in fact with some extra argument which I cannot make rigorous for the moment, this complement has Euler characteristic zero because it seems to be a fiber bundle over some base with fiber equal to circle, and then your argument works anyway. | |
| Jun 11, 2014 at 13:49 | comment | added | Reladenine Vakalwe | @semyonalesker: The way I see it is that the tubular neghborhood $T$ of $Y$ intersected with $X-Y$ (i.e, $T-Y$) is diffeomorphic to a vector bundle with its zero section removed. So basically a sphere bundle. I believe that Dan was pointing out that to get that this has Euler characteristic $0$ (as I am claiming implicitly in my argument) it needs to be an odd sphere. But this is automatic since my stratum, being a complex variety, has even real dimension. So I do indeed get an odd sphere. | |
| Jun 10, 2014 at 16:43 | comment | added | asv | @DanPetersen: I do not see that one gets sphere bundle. My impression is that it should be a bundle with odd dimensional fibers, perhaps even stratified by odd dimensional strata. If the fiber is smooth then its Euler characteristic vanishes anyway. But if it is not smooth then I am less sure. | |
| Jun 10, 2014 at 15:28 | vote | accept | asv | ||
| Jun 10, 2014 at 16:33 | |||||
| Jun 10, 2014 at 14:10 | comment | added | Reladenine Vakalwe | @DanPetersen: The strata are complex varieties, so even real dimension (am I being silly with something?). Maybe I should have said that complex varieties admit algebraic stratifications that satisfy the Whitney conditions. | |
| Jun 10, 2014 at 14:00 | comment | added | Dan Petersen | To deduce additivity of Euler characteristic you need to use also that all strata are even dimensional. For instance additivity fails for a point on the real line. The point being that the intersection of the two opens in the M-V sequence is a sphere bundle, and only odd spheres have vanishing Euler characteristic... | |
| Jun 10, 2014 at 13:53 | history | edited | Allen Knutson | CC BY-SA 3.0 |
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| Jun 10, 2014 at 13:31 | history | answered | Reladenine Vakalwe | CC BY-SA 3.0 |